Question

A variable plane remains at constant distance $p$ from the origin.If it meets coordinate axes at points $A,B,C$ then the locus of thecentroid of $\Delta ABC$ is

• A. $x^{-2}+y^{-2}+z^{-2}=9p^{-2}$
• B. $x^{-3}+y^{-3}+z^{-3}=9p^{-3}$
• C. $x^2+y^2+z^2=9p^2$
• D. $x^3+y^3+z^3=9p^3$

Answer 1

Answer: (A)

Let $A\equiv (a,0,0),B\equiv (0,b,0),C\equiv (0,0,c)$,
then equation of the plane is
$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
Its distance from the origin,
$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{p^2}\dots$ (i)
If $(x,y,z)$ be centroid of $\Delta ABC$, then
$x=\frac{a}{3},y=\frac{b}{3},z=\frac{c}{3}\dots$(ii)
Eliminating $a,b,c$ from (i) and (ii) required locus is
$x^{-2}+y^{-2}+z^{-2}=9p^{-2}$